Lie groups actions and curvilinear coordinates question

Hello,
let's suppose I have the following system of curvilinear coordinates in ℝ2: [tex]x(u,v) = u[/tex] [tex]y(u,v) = v + e^u[/tex] where one arbitrary coordinate line [itex]C_\lambda(u)=u \mathbf{e_1} + (\lambda+e^u) \mathbf{e_2}[/itex] represents the orbit of some point in ℝ2 under the action of a Lie group.

Now I consider another Lie group that produces the same orbits, but rotated by 90°, that is: [tex]K_s(t)=(e^{-t}+s) \mathbf{e_1} + t \mathbf{e_2}[/tex].

My questions are:

1) How can I find a system of curvilinear coordinates x(u,t), y(u,t) such that when keeping t fixed I would obtain the orbits C(u), and when keeping u fixed I would obtain the orbits K(t) ?

2) What are the conditions that these two Lie groups should satisfy in order for this to be possible? For example, had our orbits been paraboloids of the kind [itex]C(u)=u \mathbf{e_1} + (u^2+\lambda) \mathbf{e_2}[/itex], then no such coordinate system can exist because the other Lie group that produces the parabolae rotated by 90° will give orbits that intersects at two points with the orbits C(u) of the first Lie group